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Multipartite Entanglement Distribution in Quantum Networks using Subgraph Complementations

Aniruddha Sen, Kenneth Goodenough, Don Towsley

TL;DR

The paper tackles efficient distribution of graph states across quantum networks and addresses resource and time trade-offs under noise. It introduces subgraph complementation (SC) as a primitive that can generate any graph state via a sequence of SC operations, with the cost tied to $\mathbb{c}_2(G)$ and entanglement measures such as $\mathbb{E}_r(G)$ and the minimum rank. The authors show that for sparse graphs the distribution time scales as $O(n)$ and, with parallelization, as $O(n \log n)$ for many dense graphs; in some cases a constant-depth scheme yields $O(\log n)$, while fidelity under depolarizing noise improves relative to prior FNT and bipartite approaches. They also provide greedy methods to construct $\mathcal{SC}_{sys}$, classify graph-state families, and suggest extensions to stabilizer states, with potential impact on quantum networks and error correction.

Abstract

Quantum networks are important for quantum communication, enabling tasks such as quantum teleportation, quantum key distribution, quantum sensing, and quantum error correction, often utilizing graph states, a specific class of multipartite entangled states that can be represented by graphs. We propose a novel approach for distributing graph states across a quantum network. We show that the distribution of graph states can be characterized by a system of subgraph complementations, which we also relate to the minimum rank of the underlying graph and the degree of entanglement quantified by the Schmidt-rank of the quantum state. We analyze resource usage for our algorithm and show that it improves on the number of qubits, bits for classical communication, and EPR pairs utilized, as compared to prior work. In fact, the number of local operations and resource consumption for our approach scales linearly in the number of vertices. This produces a quadratic improvement in completion time for several classes of graph states represented by dense graphs, which translates into an exponential improvement by allowing parallelization of gate operations. This leads to improved fidelities in the presence of noisy operations, as we show through simulation in the presence of noisy operations. We classify common classes of graph states, along with their optimal distribution time using subgraph complementations. We find a sequence of subgraph complementation operations to distribute an arbitrary graph state which we conjecture is close to the optimal sequence, and establish upper bounds on distribution time along with providing approximate greedy algorithms.

Multipartite Entanglement Distribution in Quantum Networks using Subgraph Complementations

TL;DR

The paper tackles efficient distribution of graph states across quantum networks and addresses resource and time trade-offs under noise. It introduces subgraph complementation (SC) as a primitive that can generate any graph state via a sequence of SC operations, with the cost tied to and entanglement measures such as and the minimum rank. The authors show that for sparse graphs the distribution time scales as and, with parallelization, as for many dense graphs; in some cases a constant-depth scheme yields , while fidelity under depolarizing noise improves relative to prior FNT and bipartite approaches. They also provide greedy methods to construct , classify graph-state families, and suggest extensions to stabilizer states, with potential impact on quantum networks and error correction.

Abstract

Quantum networks are important for quantum communication, enabling tasks such as quantum teleportation, quantum key distribution, quantum sensing, and quantum error correction, often utilizing graph states, a specific class of multipartite entangled states that can be represented by graphs. We propose a novel approach for distributing graph states across a quantum network. We show that the distribution of graph states can be characterized by a system of subgraph complementations, which we also relate to the minimum rank of the underlying graph and the degree of entanglement quantified by the Schmidt-rank of the quantum state. We analyze resource usage for our algorithm and show that it improves on the number of qubits, bits for classical communication, and EPR pairs utilized, as compared to prior work. In fact, the number of local operations and resource consumption for our approach scales linearly in the number of vertices. This produces a quadratic improvement in completion time for several classes of graph states represented by dense graphs, which translates into an exponential improvement by allowing parallelization of gate operations. This leads to improved fidelities in the presence of noisy operations, as we show through simulation in the presence of noisy operations. We classify common classes of graph states, along with their optimal distribution time using subgraph complementations. We find a sequence of subgraph complementation operations to distribute an arbitrary graph state which we conjecture is close to the optimal sequence, and establish upper bounds on distribution time along with providing approximate greedy algorithms.
Paper Structure (24 sections, 3 theorems, 13 equations, 14 figures, 1 table)

This paper contains 24 sections, 3 theorems, 13 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Summary of Contributions
  4. Background
  5. Subgraph complementation operation
  6. Single subgraph complementation
  7. Sequence of subgraph complementations
  8. Analysis
  9. Graph State Distribution Algorithm
  10. Subgraph complementation system
  11. Time for subgraph complementation
  12. Entanglement of the graph state
  13. Detailed comparison to prior work
  14. Incorporating Parallelization
  15. Log depth GHZ state generation
  16. ...and 9 more sections

Key Result

Theorem 4.1

Given a graph $G$ with $n$ vertices and the corresponding system $\mathcal{SC}_{sys}$ with minimal size $\mathbb{c}_2(G)$ such that each $SC$ operation on $V_i$ is performed on $k_i$ vertices, then $T(G,\mathcal{SC}_{sys}) = \mathbb{c}_2(G) \cdot t(\overline{k})\,,$ where $\overline{k} = \frac{1}{d}

Figures (14)

  • Figure 1: 4-clique addition operation. The SC operation is first carried out on $V'=\{C_0,C_1,C_2\}$ containing qubits $\{c_0,c_1,c_2\}$.
  • Figure 2: Edge reset operation.
  • Figure 3: Subgraph complementation system for $W_5$Buchanan2021SubgraphComp.
  • Figure 4: Parallelized subgraph complementation. The solid lines indicate the inner edges whereas the dashed lines indicate the outer edges which were added in step (c) of the parallelized $SC$ operation.
  • Figure 5: Comparison of fidelity of a 3-qubit GHZ state distributed according to the SC algorithm (\ref{['SCalg']}), the FNT algorithm (Bipartite A protocol) (\ref{['fntalg']}), and the Bipartite B protocol, considering (left) and neglecting (right) noise from local correction unitaries, having $0\le p\le 0.015$, with the results obtained over 10000 trials. The red line indicates a fidelity of 0.75.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • proof